Externally studentized normal midrange distribution

Batista, Ben Dêivide de Oliveira; Ferreira, Daniel Furtado; Chaves, Lucas Monteiro

doi:10.1590/1413-70542017414047716

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Agricultural Sciences • Ciênc. agrotec. 41 (4) • Jul-Aug 2017 • https://doi.org/10.1590/1413-70542017414047716 copy

Externally studentized normal midrange distribution

Distribuição da midrange estudentizada externamente

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The distribution of externally studentized midrange was created based on the original studentization procedures of Student and was inspired in the distribution of the externally studentized range. The large use of the externally studentized range in multiple comparisons was also a motivation for developing this new distribution. This work aimed to derive analytic equations to distribution of the externally studentized midrange, obtaining the cumulative distribution, probability density and quantile functions and generating random values. This is a new distribution that the authors could not find any report in the literature. A second objective was to build an R package for obtaining numerically the probability density, cumulative distribution and quantile functions and make it available to the scientific community. The algorithms were proposed and implemented using Gauss-Legendre quadrature and the Newton-Raphson method in R software, resulting in the SMR package, available for download in the CRAN site. The implemented routines showed high accuracy proved by using Monte Carlo simulations and by comparing results with different number of quadrature points. Regarding to the precision to obtain the quantiles for cases where the degrees of freedom are close to 1 and the percentiles are close to 100%, it is recommended to use more than 64 quadrature points.

Index terms:
Distribution function; density function; Gauss-Legendre quadrature; Newton-Raphson method; R.

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[1] ^*Corresponding author: ben.deivide@gmail.com

Values			Method
n	v	or	Quadrature	MC
15	4	4.000	0.999712051088144	0.999719000280999
30	7	2.000	0.999521701960583	0.999520000479999
45	10	4.000	0.999999689708233	0.999999999999999
60	25	1.000	0.996635357142795	0.996665003334996
20	2	1.000	0.941476242577670	0.941947058052942
30	5	0.300	0.786876942543113	0.786846213153786
90	40	0.200	0.748082418017427	0.748314251685748
30	10	0.000	0.500000000000000	0.500382499617500
40	5	-1.000	0.016709246604515	0.016733983266017
20	20	-0.400	0.147628604257637	0.147586852413147
15	∞	0.300	0.778662390742543	0.778745221254779
60	∞	0.800	0.991576069342297	0.991477008522991

Values			Quadrature
n	v	or	s = 64 points	i = 250 points	ɛ
15	4	4.000	0.999712051088145	0.99971205062441360	4.637313 x 10^-10
30	7	2.000	0.999521701960583	0.9995217019605828	1.110223 x 10^-10
45	10	4.000	0.999999689708230	0.99999968970816311	6.694644 x 10^-10
60	25	1.000	0.996635357142795	0.99663535714279172	3.330669 x 10^-10
20	2	1.000	0.941476242577670	0.94147624257766860	1.443290 x 10^-10
30	5	0.300	0.786876942543113	0.78687694254311258	4.440892 x 10^-10
90	40	0.200	0.748082418017427	0.74808241801730846	1.185718 x 10^-10
30	10	0.000	0.500000000000002	0.50000000000000200	1.000000 x 10^-10
40	5	-1.000	0.016709246604515	0.01670924660451461	3.920475 x 10^-10
20	20	-0.400	0.147628604257637	0.14762860425763724	2.498002 x 10^-10
15	∞	0.300	0.778662390742543	0.77866239074254306	3.330669 x 10^-10
60	∞	0.800	0.991576069342296	0.99157606934229103	5.662137 x 10^-10

		Sample size (n)
v		2	3	4	5	6	7	8	9	10
1		4.464	3.799	3.445	3.219	3.060	2.941	2.846	2.770	2.705
2		2.065	1.757	1.594	1.490	1.417	1.362	1.319	1.284	1.254
3		1.664	1.416	1.285	1.201	1.143	1.098	1.064	1.035	1.012
4		1.507	1.283	1.164	1.088	1.035	0.995	0.964	0.938	0.917
5		1.425	1.213	1.100	1.029	0.978	0.941	0.911	0.887	0.867
6		1.374	1.169	1.061	0.992	0.944	0.907	0.879	0.855	0.836
7		1.340	1.140	1.034	0.967	0.920	0.885	0.857	0.934	0.815
8		1.315	1.119	1.015	0.949	0.903	0.868	0.841	0.819	0.800
9		1.296	1.103	1.001	0.936	0.890	0.856	0.829	0.807	0.789
10		1.282	1.091	0.990	0.925	0.880	0.846	0.820	0.798	0.780
15		1.240	1.055	0.957	0.895	0.851	0.819	0.793	0.772	0.754
16		1.235	1.051	0.953	0.891	0.848	0.815	0.789	0.769	0.751
17		1.230	1.047	0.950	0.888	0.845	0.812	0.787	0.766	0.748
18		1.226	1.044	0.947	0.885	0.842	0.810	0.784	0.763	0.746
19		1.223	1.041	0.944	0.883	0.840	0.807	0.782	0.761	0.744
20		1.220	1.038	0.942	0.881	0.838	0.805	0.780	0.759	0.742
25		1.208	1.028	0.933	0.872	0.830	0.798	0.772	0.752	0.735
30		1.200	1.021	0.927	0.867	0.824	0.793	0.768	0.747	0.730
50		1.185	1.009	0.915	0.856	0.814	0.783	0.758	0.738	0.721
100		1.174	0.999	0.906	0.848	0.806	0.775	0.751	0.731	0.714
150		1.170	0996	0.904	0.845	0.804	0.773	0.748	0.729	0.712
200		1.168	0.994	0.902	0.844	0.803	0.772	0.747	0.727	0.711
300		1.167	0.993	0.901	0.842	0.801	0.770	0.746	0.726	0.710
1000		1.164	0.991	0.899	0.841	0.800	0.769	0.744	0.725	0.708
∞		1.163	0.990	0.898	0.840	0.799	0.768	0.744	0.724	0.708
	Sample size (n)
v	15	20	25	30	35	0.799	0.768	0.744	0.724	0.708	40	45	50	75	100
1	2.492	2.366	2.280	2.215	2.165	2.124	2.090	2.060	1.957	1.893
2	1.156	1.099	1.059	1.030	1.007	0.988	0.972	0.959	0.912	0.882
3	0.933	0.887	0.855	0.832	0.813	0.798	0.785	0.774	0.736	0.713
4	0.846	0.804	0.775	0.754	0.737	0.723	0.712	0.702	0.668	0.646
5	0.800	0.760	0.733	0.713	0.697	0.684	0.673	0.664	0.632	0.611
6	0.771	0.733	0.707	0.687	0.672	0.660	0.649	0.640	0.609	0.590
7	0.752	0.715	0.689	0.670	0.656	0.643	0.633	0.625	0.594	0.575
8	0.738	0.702	0.677	0.658	0.643	0.632	0.622	0.613	0.583	0.564
9	0.728	0.692	0.667	0.649	0.634	0.623	0.613	0.604	0.575	0.556
10	0.719	0.684	0.660	0.641	0.627	0.616	0.606	0.598	0.568	0.550
15	0.696	0.661	0.638	0.620	0.607	0.596	0.586	0.578	0.550	0.532
16	0.693	0.659	0.635	0.618	0.604	0.593	0.584	0.576	0.548	0.530
17	0.691	0.656	0.633	0.616	0.602	0.591	0.582	0.574	0.546	0.528
18	0.688	0.645	0.631	0.614	0.600	0.589	0.580	0.572	0.544	0.527
19	0.686	0.652	0.629	0.612	0.598	0.587	0.578	0.570	0.542	0.525
20	0.685	0.651	0.628	0.610	0.597	0.586	0.577	0.569	0.541	0.524
25	0.678	0.645	0.622	0.605	0.591	0.580	0.571	0.563	0.536	0.519
30	0.674	0.640	0.618	0.601	0.587	0.577	0.568	0.560	0.532	0.515
50	0.665	0.632	0.610	0.593	0.580	0.569	0.560	0.553	0.526	0.509
100	0.659	0.626	0.604	0.588	0.575	0.564	0.555	0.548	0.521	0.504
150	0.657	0.625	0.602	0.586	0.573	0.562	0.553	0.546	0.519	0.503
200	0.656	0.624	0.601	0.585	0.572	0.561	0.553	0.545	0.518	0.502
300	0.655	0.623	0.601	0.584	0.571	0.561	0.552	0.544	0.518	0.501
1000	0.654	0.621	0.599	0.583	0.570	0.559	0.551	0.543	0.517	0.500
∞	0.653	0.621	0.599	0.582	0.569	0.559	0.550	0.543	0.516	0.500